Pricing financial instruments, e.g., financial derivatives, is a complex art requiring substantial expertise and experience. Trading financial instruments, such as options, involves a sophisticated process of pricing typically performed by a trader.
The term “option” in the context of the present application is defined broadly as any financial instrument having option-like properties, e.g., any financial derivative including an option or an option-like component. This category of financial instruments may include any type of option or option-like financial instrument, relating to some underlying asset Assets as used in this application include anything of value; tangible or non-tangible, financial or non-financial. For example, as used herein, options range from a simple Vanilla option on a single stock and up to complex convertible bonds whose convertibility depends on some key, e.g., the weather.
The price of an asset for immediate (e.g., 2 business days) delivery is called the spot price. For an asset sold in an option contract, the strike price is the agreed upon price at which the deal is executed if the option is exercised. For example, a foreign exchange (FX) option involves buying or selling an amount of one currency for an amount of another currency. The spot price is the current exchange rate between the two currencies on the open market. The strike price is the agreed upon exchange rate of the currency if the option is exercised.
To facilitate trading of options and other financial instruments, a trader prepares a bid price and offer price (also called ask price) for a certain option. The bid price is the price at which the trader is willing to purchase the option and the offer price is the price at which the trader is willing to sell the option. When another trader is interested in the option the first trader quotes both the bid and offer prices, not knowing whether the second trader is interested in selling or buying. The offer price is higher than the bid price and the difference between the offer and bid is referred to as bid-offer spread.
A call option is an option to buy an asset at a certain price (i.e., a strike price) on a certain date. A put option is an option to sell an asset at a strike price on a certain date. At any time prior to the option expiration date, the holder of the option may determine whether or not to exercise the option, depending on the current exchange rate (spot) for that currency. If the spot (i.e., current market) price is lower than the strike price, the holder may choose not to exercise the call option and lose only the cost of the option itself. However, if the strike is lower than the spot, the holder may exercise the right to buy the currency at the strike price making a profit equal to the difference between the spot and the strike prices.
A forward rate is the future exchange rate of an asset at a given future day on which the exchange transaction is performed based on an option contract. The forward rate is calculated based on a current rate of the asset, a current interest rate prevailing in the market, expected dividends (for stocks), cost of carry (for commodities), and other parameters depending on the underlying asset of the option.
An at-the-money forward option (ATM) is an option whose strike is equal to the forward rate of the asset. In this application, at-the-money forward options are generically referred to as at-the-money options, as is the common terminology in the foreign exchange (FX) and other financial markets. An in-the-money call option is a call option whose strike is below the forward rate of the underlying asset, and an in-the-money put option is a put option whose strike is above the forward rate of the underlying asset. An out-of-the-money call option is a call option whose strike is above the forward rate of the underlying asset, and an out-of-the-money put option is a put option whose strike is below the forward rate of the underlying asset.
An exotic option, in the context of this application, is a generic name referring to any type of option other than a standard Vanilla option. While certain types of exotic options have been extensively and frequently traded over the years, and are still traded today, other types of exotic options had been used in the past but are no longer in use today. Currently, the most common exotic options include are “barrier” options, “binary” options, “digital” options, “partial barrier” options (also known as “window” options), “average” options and “quanto” options. Some exotic options can be described as a complex version of the standard (Vanilla) option. For example, barrier options are exotic options where the payoff depends on whether the underlying asset's price reaches a certain level, hereinafter referred to as “rigger”, during a certain period of time. The “pay off” of an option is defined as the cash realized by the holder of the option upon its expiration. There are generally two types of barrier options, namely, a knock-out option and a knock-in option. A knock-out option is an option that terminates if and when the spot reaches the trigger. A knock-in option comes into existence only when the underlying asset's price reaches the trigger. It is noted that the combined effect of a knock-out option with strike K and trigger B and a knock-in option with strike K and trigger B, both having the same expiration, is equivalent to a corresponding Vanilla option with strike K. Thus, knock-in options can be priced by pricing corresponding knock-out and vanilla options. Similarly, a one-touch option can be decomposed into two knock-in call options and two knock-in put options, a double one-touch option can be decomposed into two double knock-out options, and so on. It is appreciated that there are many other types of exotic options known in the art.
Certain types of options, e.g., Vanilla options, are commonly categorized as either European or American. A European option can be exercised only upon its expiration. An American option can be exercised at any time after purchase and before expiration. For example, an American Vanilla option has all the properties of the Vanilla option type described above, with the additional property that the owner can exercise the option at any time up to and including the option's expiration date. As is known in the art, the right to exercise an American option prior to expiration makes American options more expensive than corresponding European options. Generally in this application, the term “Vanilla” refers to a European style Vanilla option. European Vanilla options are the most commonly traded options; they are traded both on exchanges and over the counter (OTC). The much less common American Vanilla options are traded exclusively OTC, and are difficult to price.
U.S. Pat. No. 5,557,517 (“the '517 patent”) describes a method of pricing American Vanilla options for trading in a certain exchange. This patent describes a method of pricing Call and Put American Vanilla options, where the price of the option depends on a constant margin or commission required by the market maker. The method of the '517 patent ignores data that may affect the price of the option, except for the current price of the underlying asset and, thus, this method can lead to serious errors, for example, an absurd result of a negative option price. Clearly, this method does not emulate the way American style Vanilla options are priced in real markets.
The Black-Scholes model (developed in 1975) is a widely accepted method for valuing options. This model calculates a probability-based theoretical value (TV), which is commonly used as a starting point for approximating option prices. This model is based on a presumption that the change in the rate of the asset generally follows a Brownian motion, as is known in the art. Using such Brownian motion model, known also as a stochastic process, one may calculate the theoretical price of any type of financial derivative, either analytically, as is the case for the exotic options discussed above, or numerically. For example, it is common to calculate the theoretical price of complicated financial derivatives through simulation techniques, such as the Monte Carlo method, introduced by Boyle in 1977. Such techniques may be useful in calculating the theoretical value of an option, provided that the computer being used is sufficiently powerful to handle all the calculations involved. In the simulation method, the computer generates many propagation paths for the underlying asset, starting at the trade time and ending at the time of the option expiry. Each path is discrete and generally follows the Brownian motion probability, but may be generated as densely as necessary by reducing the time lapse between each move of the underlying asset. Thus, if the option is path-dependant, each path is followed and only the paths that satisfy the conditions of the option are taken into account. The end results of each such path are summarized and lead to the theoretical price of the derivative.
The original Black Scholes model is designed for calculating theoretical prices for Vanilla options. However, it should be understood that any reference in this application to the Black-Scholes model refers to use of any model known in the art for calculating theoretical prices of options, e.g., a Brownian motion model, as applied to any type of option, including exotic options. Furthermore, this application is general and independent of the way in which the theoretical value of the option is obtained. It can be derived analytically, numerically, using any kind of simulation method or any other technique available.
For example, U.S. Pat. No. 6,061,662 (“the '662 patent”) describes a method of evaluating the theoretical price of an option using a Monte Carlo method based on historical data. The simulation method of the '662 patent uses stochastic historical data with a predetermined distribution function in order to evaluate the theoretical price of options. Examples is the '662 patent are used to illustrate that this method generates results which are very similar to those obtained by applying the Black-Scholes model to Vanilla options. Unfortunately, methods based on historical data alone are not relevant for simulating financial markets, even for the purpose of theoretical valuation. For example, one of the most important parameters used for valuation of options is the volatility of the underlying asset, which is a measure for how the rate of the underlying asset fluctuates. It is well known that the financial markets use predicted, or “future”, value for the volatility of the underlying assets, which often deviates dramatically from the historical data. In market terms, future volatility is often referred to as “implied volatility”, and is differentiated from “historical volatility”. For example, the implied volatility tends to be much higher than the historical volatility of the underlying asset before a major event, such as risk of war, or during and after a financial crisis.
It is appreciated by persons skilled in the art that the Black-Scholes model is a limited approximation that may yield results very far from real market prices and, thus, corrections to the Black-Scholes model must generally be added by traders. In the foreign exchange (FX) Vanilla market, for example, the market trades in volatility terms and the translation to option price is performed through use of the Black-Scholes formula. In fact, traders commonly refer to using the Black-Scholes model as “using the wrong volatility with the wrong model to get the right price”.
In order to adjust the price, in the Vanilla market, traders use different volatilities for different strikes, i.e., instead of using one volatility per asset, a trader may use different volatility values for a given asset depending on the strike price. This adjustment is known as volatility “smile” adjustment. The origin of the term “smile”, in this context, is in the foreign exchange market, where the volatility of a commodity becomes higher as the commodity's price moves further away from the ATM strike.
The phrase “market price of a derivative” is used herein to distinguish between the single value produced by some benchmark models, such as the Black-Scholes model, and the actual bid and offer prices traded in the real market. For example, in some options, the market bid side may be twice the Black-Scholes model price and the offer side may be three times the Black-Scholes model price.
Many exotic options are characterized by discontinuity of the payout and, therefore, a discontinuity in some of the risk parameters near the trigger(s). This discontinuity prevents an oversimplified model such as the Black-Scholes model from taking into account the difficulty in risk-managing the option. Furthermore, due to the peculiar profile of some exotic options, there may be significant transaction costs associated with re-hedging some of the risk factors. Existing models, such as the Black-Scholes model, completely ignore such risk factors.
Many factors may be taken into account in calculating option prices and corrections. (Factor-is used herein broadly as any quantifiable or computable value relating to the subject option.) Some of the notable factors are defined as follows:
Volatility (“Vol”) is a measure of the fluctuation of the return realized on an asset. An indication of the level of the volatility can be obtained by the volatility history, i.e., the standard deviation of the return of the assets for a certain past period. However, the markets trade based on a volatility that reflects the market expectations of the standard deviation in the future. The volatility reflecting market expectations is called implied volatility. In order to buy/sell volatility one commonly trades Vanilla options. For example, in the foreign exchange market, the implied volatilities of ATM Vanilla options for many frequently used option dates and currency pairs are available to users in real-time, e.g., via screens such as REUTERS, Bloomberg, TELERATE, Cantor Fitzgerald, or directly from FX option brokers.
Volatility smile, as discussed above, relates to the behavior of the implied volatility with respect to the strike, i.e., the implied volatility as a function of the strike, where the implied volatility for the ATM strike is the given ATM volatility in the market. For example, for currency options, a plot of the implied volatility as a function of the strike shows a minimum in the vicinity of the ATM strike that looks like a smile. For equity options, as another example, the volatility plot tends to be monotonous.
Vega is the rate of change in the price of an option or other derivative in response to changes in volatility, i.e., the partial derivative of the option price with respect to the volatility.
Convexity is the second partial derivative of the price with respect to the volatility, i.e. the derivative of the Vega with respect to the volatility, denoted dVega/dVol.
Delta is the rate of change in the price of an option in response to changes in the price of the underlying asset; in other words, it is a partial derivative of the option price with respect to the spot For example, a 25 delta call option is defined as follows: if against buying the option on one unit of the underlying asset, 0.25 unit of the underlying asset are sold, then for small changes in the underlying option, assuming all other factors are unchanged, the total change in the price of the option and the 0.25 unit of the asset are null.
Intrinsic value (IV) for in-the-money knock-out/knock-in exotic options with strike K and trigger (or barrier) B, is defined as IV=|B−K|/B. In-the-money knock-out/knock-in options are also referred to as Reverse knock-out/knock-in options, respectively. For a call option, the intrinsic value is the greater of the excess of the asset price over the strike price and zero. In other words, the intrinsic value of in-the-money knock out options is the intrinsic value of a corresponding Vanilla at the barrier, and represents the level of payout discontinuity in the vicinity of the trigger.
25Δ Risk Reversal (RR) is the difference between the implied volatility of a call option and a put option with the same delta (in opposite directions). Traders in the currency options market generally use 25 delta RR, which is the difference between the implied volatility of a 25 delta call option and a 25 delta put option. Thus, 25 delta RR is calculated as follows:25 delta RR=implied Vol(25 delta call)−implied Vol(25 delta put)The 25 delta risk reversal is characterized by a slope of Vega with respect to spot but practically no convexity at the current spot. Therefore it is used to price the slope dVega/dspot.
25Δ Strangle is the average of the implied volatility of the call and the put, which usually have the same delta. For example:25 delta strangle=0.5 (implied Vol (25 delta call)+implied Vol (25 delta put))The 25 delta strangle is characterized by practically no slope of Vega with respect to spot at the current spot, but a lot of convexity. Therefore it is used to price convexity. Since the at-the-money Vol is always known, it is more common to quote the butterfly in which one buys one unit of the strangle and sells 2 units of the ATM option. Like the strangle, butterfly is also quoted in volatility. For example:25 delta butterfly=0.5(implied Vol (25delta call)+implied Vol (25 delta put))−ATM VolThe reason it is more common to quote the butterfly is that butterfly provides a strategy with almost no Vega but significant convexity. Since butterfly and strangle are related through the ATM volatility, which is always known, they may be used interchangeably. The 25 delta put and the 25 delta call can be determined based on the 25 delta RR and the 25 delta strangle.
Gearing, also referred to as leverage, is the difference in price between the exotic option with the barrier and a corresponding Vanilla option having the same strike. It should be noted that a Vanilla option is always more expensive than a corresponding exotic option.
Bid/offer spread is the difference between the bid price and the offer price of a financial derivative. In the case of options, the bid/offer spread is expressed either in terms of volatility or in terms of the price of the option. The bid/offer spread of a given option depends on the specific parameters of the option. In general, the more difficult it is to manage the risk of an option, the wider is the bid/offer spread for that option.
Typically traders try to calculate the price at which they would like to buy an option (i.e., the bid side) and the price at which they would like to sell the option (i.e., the offer side). Currently, there are no mathematical or computational methods for calculating bid/offer prices, and so traders typically rely on intuition, experiments involving changing the factors of an option to see how they affect the market price, and past experience, which is considered to be the most important tool of traders. Factors commonly relied upon by traders include convexity and RR which reflect intuition on how an option should be priced. One dilemma commonly faced by traders is how wide the bid/offer spread should be. Providing too wide a spread reduces the ability to compete in the options market and is considered unprofessional, yet too narrow a spread may result in losses to the trader. In determining what prices to provide, traders need to ensure that the bid/offer spread is appropriate. This is part of the pricing process, i.e., after the trader decides where to place the bid and offer prices, he/she needs to consider whether the resultant spread is appropriate. If the spread is not appropriate, the trader needs to change either or both of the bid and offer prices in order to show the appropriate spread.